Theorem nfrexd

Description: Deduction version of nfrex . (Contributed by Mario Carneiro, 14-Oct-2016) Add disjoint variable condition to avoid ax-13 . See nfrexdg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

	Ref	Expression
Hypotheses	nfrexd.1	⊢ Ⅎ 𝑦 𝜑
	nfrexd.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
	nfrexd.3	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
Assertion	nfrexd	⊢ ( 𝜑 → Ⅎ 𝑥 ∃ 𝑦 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		nfrexd.1	⊢ Ⅎ 𝑦 𝜑
2		nfrexd.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
3		nfrexd.3	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
4		dfrex2	⊢ ( ∃ 𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀ 𝑦 ∈ 𝐴 ¬ 𝜓 )
5	3	nfnd	⊢ ( 𝜑 → Ⅎ 𝑥 ¬ 𝜓 )
6	1 2 5	nfraldw	⊢ ( 𝜑 → Ⅎ 𝑥 ∀ 𝑦 ∈ 𝐴 ¬ 𝜓 )
7	6	nfnd	⊢ ( 𝜑 → Ⅎ 𝑥 ¬ ∀ 𝑦 ∈ 𝐴 ¬ 𝜓 )
8	4 7	nfxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 ∃ 𝑦 ∈ 𝐴 𝜓 )